Differential-Difference Equations. By Richard Bellman and Kenneth L. Cooke. Pp. xvi, 465. 1963. (Academic Press)

Most of the signal processing that we will study in this course involves local operations on a signal, namely transforming the signal by applying linear combinations of values in the neighborhood of each sample point. You are familiar with such operations from Calculus, namely, taking derivatives and you are also familiar with this from optics namely blurring a signal. We will be looking at sampled signals only. Let's start with a few basic examples. Local difference Suppose we have a 1D image and we take the local difference of intensities, DI(x) = 1 2 (I(x + 1) − I(x − 1)) which give a discrete approximation to a partial derivative. (We compute this for each x in the image.) What is the effect of such a transformation? One key idea is that such a derivative would be useful for marking positions where the intensity changes. Such a change is called an edge. It is important to detect edges in images because they often mark locations at which object properties change. These can include changes in illumination along a surface due to a shadow boundary, or a material (pigment) change, or a change in depth as when one object ends and another begins. The computational problem of finding intensity edges in images is called edge detection. We could look for positions at which DI(x) has a large negative or positive value. Large positive values indicate an edge that goes from low to high intensity, and large negative values indicate an edge that goes from high to low intensity. Example Suppose the image consists of a single (slightly sloped) edge:

http://ieeexplore.ieee.org/iel5/5/31307/01456462.pdf

Linear systems

https://hal.archives-ouvertes.fr/hal-00855159/document

Wirtinger-based integral inequality: Application to time-delay systems

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Electronic Transport In Mesoscopic Systems

Technical communique: Delay-range-dependent stability for systems with time-varying delay

Preface List of symbols List of abbreviations Part I. Stability Analysis of Linear Time-Delay Systems: 1. Spectral properties of linear time-delay systems 2. Pseudospectra and robust stability analysis 3. Computation of stability regions in parameter spaces 4. Stability regions in delay-parameter spaces 5. Delays ratio sensitivity and interference 6. Stability of linear periodic systems with delays Part II. Stabilization and Robust Stabilization: 7. The continuous pole placement method 8. Stabilizability with delayed feedback: a numerical case-study 9. The robust stabilization problem 10. Stabilization using a direct eigenvalue optimization approach Part III. Applications: 11. Output feedback stabilization: the single delay case 12. Output feedback stabilization: the multiple delay case 13. Congestion control algorithms in networks 14. Smith predictor for stable systems: delay sensitivity analysis 15. Controlling unstable systems using finite spectrum assignment 16. Consensus problems in traffic flow applications 17. Stability analysis of delay models in biosciences A. Appendices Bibliography Index.

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Stability and Stabilization of Time-Delay Systems: An Eigenvalue-Based Approach

Time-delays are important components of many dynamical systems that describe coupling or interconnection between dynamics, propagation, or transport phenomena in shared environments, in heredity, and in competition in population dynamics. This monograph addresses the problem of stability analysis and the stabilisation of dynamical systems subjected to time-delays. It presents a wide and self-contained panorama of analytical methods and computational algorithms using a unified eigenvalue-based approach illustrated by examples and applications in electrical and mechanical engineering, biology, and complex network analysis.

Stability and Stabilization of Systems with Time Delay

The instability mechanisms, related to the implementation of distributed delay controllers in the context of finite spectrum assignment, were studied in detail in the past few years. In this note we introduce a distributed delay control law that assigns a finite closed-loop spectrum and whose implementation with a sum of point-wise delays is safe. This property is obtained by implicitly including a low-pass filter in the control loop. This leads to a closed-loop characteristic quasipolynomial of retarded type, and not one of neutral type, which was shown to be a cause of instability in previous schemes.

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Wim Michiels

Papers

Stability and Stabilization of Time-Delay Systems: An Eigenvalue-Based Approach

Stability and Stabilization of Systems with Time Delay

Finite spectrum assignment of unstable time-delay systems with a safe implementation

Stability and stabilization of time-delay systems

Continuous pole placement for delay equations